Theorem eulerthlem1 12365

Description: Lemma for eulerth 12371. (Contributed by Mario Carneiro, 8-May-2015.)

Hypotheses

Ref	Expression
eulerthlem1.1	|- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )
eulerthlem1.2	|- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }
eulerthlem1.3	|- T = ( 1 ... ( phi ` N ) )
eulerthlem1.4	|- ( ph -> F : T -1-1-onto-> S )
eulerthlem1.5	|- G = ( x e. T |-> ( ( A x. ( F ` x ) ) mod N ) )

Assertion

Ref	Expression
eulerthlem1	|- ( ph -> G : T --> S )

Distinct variable groups:   x, y, A   x, F, y   x, G, y   x, N, y   x, S   ph, x, y   x, T, y

Allowed substitution hint:   S( y)

Proof of Theorem eulerthlem1

Step	Hyp	Ref	Expression
1		eulerthlem1.1	. . . . . . 7 |- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )
2	1	simp2d 1012	. . . . . 6 |- ( ph -> A e. ZZ )
3	2	adantr 276	. . . . 5 |- ( ( ph /\ x e. T ) -> A e. ZZ )
4		eulerthlem1.4	. . . . . . . . . 10 |- ( ph -> F : T -1-1-onto-> S )
5		f1of 5500	. . . . . . . . . 10 |- ( F : T -1-1-onto-> S -> F : T --> S )
6	4, 5	syl 14	. . . . . . . . 9 |- ( ph -> F : T --> S )
7	6	ffvelcdmda 5693	. . . . . . . 8 |- ( ( ph /\ x e. T ) -> ( F ` x ) e. S )
8		oveq1 5925	. . . . . . . . . 10 |- ( y = ( F ` x ) -> ( y gcd N ) = ( ( F ` x ) gcd N ) )
9	8	eqeq1d 2202	. . . . . . . . 9 |- ( y = ( F ` x ) -> ( ( y gcd N ) = 1 <-> ( ( F ` x ) gcd N ) = 1 ) )
10		eulerthlem1.2	. . . . . . . . 9 |- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }
11	9, 10	elrab2 2919	. . . . . . . 8 |- ( ( F ` x ) e. S <-> ( ( F ` x ) e. ( 0..^ N ) /\ ( ( F ` x ) gcd N ) = 1 ) )
12	7, 11	sylib 122	. . . . . . 7 |- ( ( ph /\ x e. T ) -> ( ( F ` x ) e. ( 0..^ N ) /\ ( ( F ` x ) gcd N ) = 1 ) )
13	12	simpld 112	. . . . . 6 |- ( ( ph /\ x e. T ) -> ( F ` x ) e. ( 0..^ N ) )
14		elfzoelz 10213	. . . . . 6 |- ( ( F ` x ) e. ( 0..^ N ) -> ( F ` x ) e. ZZ )
15	13, 14	syl 14	. . . . 5 |- ( ( ph /\ x e. T ) -> ( F ` x ) e. ZZ )
16	3, 15	zmulcld 9445	. . . 4 |- ( ( ph /\ x e. T ) -> ( A x. ( F ` x ) ) e. ZZ )
17	1	simp1d 1011	. . . . 5 |- ( ph -> N e. NN )
18	17	adantr 276	. . . 4 |- ( ( ph /\ x e. T ) -> N e. NN )
19		zmodfzo 10418	. . . 4 |- ( ( ( A x. ( F ` x ) ) e. ZZ /\ N e. NN ) -> ( ( A x. ( F ` x ) ) mod N ) e. ( 0..^ N ) )
20	16, 18, 19	syl2anc 411	. . 3 |- ( ( ph /\ x e. T ) -> ( ( A x. ( F ` x ) ) mod N ) e. ( 0..^ N ) )
21		modgcd 12128	. . . . 5 |- ( ( ( A x. ( F ` x ) ) e. ZZ /\ N e. NN ) -> ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = ( ( A x. ( F ` x ) ) gcd N ) )
22	16, 18, 21	syl2anc 411	. . . 4 |- ( ( ph /\ x e. T ) -> ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = ( ( A x. ( F ` x ) ) gcd N ) )
23	17	nnzd 9438	. . . . . 6 |- ( ph -> N e. ZZ )
24	23	adantr 276	. . . . 5 |- ( ( ph /\ x e. T ) -> N e. ZZ )
25	16, 24	gcdcomd 12111	. . . 4 |- ( ( ph /\ x e. T ) -> ( ( A x. ( F ` x ) ) gcd N ) = ( N gcd ( A x. ( F ` x ) ) ) )
26	23, 2	gcdcomd 12111	. . . . . . 7 |- ( ph -> ( N gcd A ) = ( A gcd N ) )
27	1	simp3d 1013	. . . . . . 7 |- ( ph -> ( A gcd N ) = 1 )
28	26, 27	eqtrd 2226	. . . . . 6 |- ( ph -> ( N gcd A ) = 1 )
29	28	adantr 276	. . . . 5 |- ( ( ph /\ x e. T ) -> ( N gcd A ) = 1 )
30	24, 15	gcdcomd 12111	. . . . . 6 |- ( ( ph /\ x e. T ) -> ( N gcd ( F ` x ) ) = ( ( F ` x ) gcd N ) )
31	12	simprd 114	. . . . . 6 |- ( ( ph /\ x e. T ) -> ( ( F ` x ) gcd N ) = 1 )
32	30, 31	eqtrd 2226	. . . . 5 |- ( ( ph /\ x e. T ) -> ( N gcd ( F ` x ) ) = 1 )
33		rpmul 12236	. . . . . 6 |- ( ( N e. ZZ /\ A e. ZZ /\ ( F ` x ) e. ZZ ) -> ( ( ( N gcd A ) = 1 /\ ( N gcd ( F ` x ) ) = 1 ) -> ( N gcd ( A x. ( F ` x ) ) ) = 1 ) )
34	24, 3, 15, 33	syl3anc 1249	. . . . 5 |- ( ( ph /\ x e. T ) -> ( ( ( N gcd A ) = 1 /\ ( N gcd ( F ` x ) ) = 1 ) -> ( N gcd ( A x. ( F ` x ) ) ) = 1 ) )
35	29, 32, 34	mp2and 433	. . . 4 |- ( ( ph /\ x e. T ) -> ( N gcd ( A x. ( F ` x ) ) ) = 1 )
36	22, 25, 35	3eqtrd 2230	. . 3 |- ( ( ph /\ x e. T ) -> ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = 1 )
37		oveq1 5925	. . . . 5 |- ( y = ( ( A x. ( F ` x ) ) mod N ) -> ( y gcd N ) = ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) )
38	37	eqeq1d 2202	. . . 4 |- ( y = ( ( A x. ( F ` x ) ) mod N ) -> ( ( y gcd N ) = 1 <-> ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = 1 ) )
39	38, 10	elrab2 2919	. . 3 |- ( ( ( A x. ( F ` x ) ) mod N ) e. S <-> ( ( ( A x. ( F ` x ) ) mod N ) e. ( 0..^ N ) /\ ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = 1 ) )
40	20, 36, 39	sylanbrc 417	. 2 |- ( ( ph /\ x e. T ) -> ( ( A x. ( F ` x ) ) mod N ) e. S )
41		eulerthlem1.5	. 2 |- G = ( x e. T |-> ( ( A x. ( F ` x ) ) mod N ) )
42	40, 41	fmptd 5712	1 |- ( ph -> G : T --> S )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2164   {crab 2476   |-> cmpt 4090   -->wf 5250   -1-1-onto->wf1o 5253   ` cfv 5254  (class class class)co 5918   0cc0 7872   1c1 7873   x. cmul 7877   NNcn 8982   ZZcz 9317   ...cfz 10074  ..^cfzo 10208   mod cmo 10393   gcd cgcd 12079   phicphi 12347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-sup 7043  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-dvds 11931  df-gcd 12080

This theorem is referenced by:  eulerthlemh  12369  eulerthlemth  12370